# Alternating group:A5

From Groupprops

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this groupView a complete list of particular groups (this is a very huge list!)[SHOW MORE]

*This particular group is the smallest (in terms of order):* simple non-Abelian group

## Contents

## Definition

The alternating group is defined in the following ways:

- It is the group of even permutations (viz, the alternating group) on five elements
- It is the von Dyck group (sometimes termed triangle group) with parameters
- It is the group of orientation-preserving symmetries of the regular icosahedron (or equivalently, regular dodecahedron)
- It is the projective special linear group of order two over the field of five elements, viz

## Group properties

## Subgroups

## Endomorphisms

## Bigger groups

### Groups having it as a subgroup

The alternating group is a subgroup of index two inside the symmetric group on five elements. It is also of index two in the full icosahedral symmetry group, which turns out *not* to be , but instead the direct product of and the cyclic group of order two.

### Groups having it as a quotient

The alternating group is a quotient of by its center. Hence, it is the inner automorphism group of . is also the universal central extension of the alternating group.